This thesis aims to prove instance optimality of adaptive finite element methods (AFEMs) for various model problems. Based on the concept of populations, which allows for the proof of certain geometric properties of meshes and their sets of nodes, as well as the concept of energy, which is closely related to the finite element solution of a mesh and its approximation error, an abstract framework is developed for proving instance optimality of an AFEM for selected problems. Three properties will turn out to be sufficient for instance optimality: a lower diamond estimate of the energy, discrete local equivalence of energy and a posteriori error estimator, as well as an assumption on the marking step. These properties will be shown to be valid for two model problems. First, elliptic diffusion problems with mixed Neumann and homogeneous Dirichlet boundary conditions will be considered, which are discretised by conforming finite elements of arbitrary order. Furthermore, the abstract framework will be applied to goal oriented adaptive finite element methods (GOAFEM), in which the quantity of interest is the value of a linear functional of the solution. To estimate the error of this value, a modified error quantity is introduced, for which instance optimality is shown. Finally, the theoretical findings are underpinned by numerical experiments. Overall, the present diploma thesis generalizes the work [Diening, Kreuzer, Stevenson; Found. Comput. Math. 16 (2016)], which proves instance optimality of an adaptive P1-FEM for the Poisson problem with homogeneous Dirichlet boundary conditions.